GW approximation with $\text{LSDA}+\text{U}$ method and applications to NiO, MnO, and ${\text{V}}_2{\text{O}}_3$

A GW approximation (GWA) method named $\text{U}+\text{GWA}$ is proposed, where we can start GWA with more localized wave functions obtained by the local spin-density approximation $\left(\text{LSDA}\right)+\text{U}$ method. Then GWA and $\text{U}+\text{GWA}$ are applied to MnO, NiO, and ${\text{V}}_2{\text{O}}_3$ in antiferromagnetic phase. The band gaps and energy spectra show excellent agreement with the experimentally observed results and are discussed in detail. The calculated width of $d$ bands of ${\text{V}}_2{\text{O}}_3$ is much narrower than that of the observed one which may be a mixture of $t_{2g}^2$ multiplet and single-electron $t_{2g}$ level. GWA or $\text{U}+\text{GWA}$ does not work also in the paramagnetic phase of ${\text{V}}_2{\text{O}}_3$ and the reason for this is clarified. The method of the unique choice of on-site Coulomb interaction is discussed in detail. The criterion for whether we should adopt GWA or $\text{U}+\text{GWA}$ is discussed and is assessed with the help of the off-diagonal elements of the self-energy.

Figure 1

(Color online) (a) Quasiparticle energy bands along symmetry lines [$\Gamma =\left(0,0,0\right)$, $\text{X}=\frac{2\pi }{a}\left(1,0,0\right)$, $\text{K}=\frac{2\pi }{a}\left(\frac{3}{4},\frac{3}{4},0\right)$, and point $\frac{2\pi }{a}\left(1,1,0\right)$ equivalent to $X$ point] and (b) quasiparticle density of states of antiferromagnetic MnO. Solid and broken lines refer to those by GWA and those by LSDA. The energy zeroth is fixed at the top of the valence band.

Figure 2

(Color online) Imaginary part of Green’s functions $\left(1/\pi \right)\left|\text{Im}G\left(E\right)\right|$ of antiferromagnetic MnO by GWA. (a) The total Green’s function (solid curve) and experimental XPS and BIS spectra (dotted line) (Ref. 15). (b) The partial Green’s function per atom. The energy zeroth is fixed at the top of the valence band.

Figure 3

(Color online) (a) Quasiparticle energy bands and (b) quasiparticle density of states of antiferromagnetic NiO. Symmetry points and lines are the same as those in Fig. 1. Solid and broken lines refer to those by $\text{U}+\text{GWA}$ and $\text{LSDA}+\text{U}$ with $U=2.5\text{eV}$ and $J=0.95\text{eV}$, respectively. In (a), the cross marks refer to the result by the angle-resolved photoemission spectra (Ref. 19). The energy zeroth is fixed at the top of the valence band.

Figure 4

(Color online) Imaginary part of Green’s functions $\left(1/\pi \right)\left|\text{Im}G\left(E\right)\right|$ of antiferromagnetic NiO by $\text{U}+\text{GWA}$ with $U=2.5\text{eV}$ and $J=0.95\text{eV}$. (a) The total Green’s function (solid line) and experimental XPS and BIS spectra (dotted line) (Ref. 18). (b) The partial Green’s function per atom. The energy zeroth is fixed at the top of the valence band.

Figure 5

(Color online) $U$ dependence of the static screened Coulomb interaction $W_U$. $\times$: $\text{LSDA}+\text{U}$; open square $◻$: $\text{U}+\text{GWA}$; broken line: Eq. (25); and closed square: Eq. (27) for MnO. $+$: $\text{LSDA}+\text{U}$; open circle $○$: $\text{U}+\text{GWA}$; solid line: Eq. (25); closed circle: Eq. (27) for NiO. See Sec. 3C for the explanation on the two solutions of $\text{U}+\text{GWA}$ $(○)$ at $U=1$ and $U=2$ in NiO.

Figure 6

(Color online) The value ${\left(\sqrt{x^2+1}-x\right)}^2$ is shown, whose definition should be referred to in the text and Eq. (28). The vertical and horizontal axes are the LSDA eigenenergies for GWA or $\text{LSDA}+\text{U}$ eigenenergies $\left(\text{U}+\text{GWA}\right)$ ${ϵ}_{\mathbf{k}n}$ and ${ϵ}_{\mathbf{k}{n}^{\prime }}$. The energy zeroth is set at the Fermi energy. The area having larger values in GWA disappears in $\text{U}+\text{GWA}$ of NiO and no area having larger values exists in GWA of MnO.

Figure 7

(Color online) (a) Spin structure of low-temperature antiferromagnetic insulator phase of ${\text{V}}_2{\text{O}}_3$. Arrows stand for the spin directions of V ions. The Greek letters $\alpha$, $\beta$, $\gamma$, and $\delta$ denote the first-, second-, third-, and fourth-nearest-neighbor $V\text{−}V$ pairs, respectively. The vectors $\mathbf{a}$, $\mathbf{b}$, and their perpendicular vector $\mathbf{c}$ are the primitive vectors of the hexagonal unit cell, and the vectors ${\mathbf{a}}_m$, ${\mathbf{b}}_m$, and ${\mathbf{c}}_m$ are those of the monoclinic cell of antiferromagnetic phase. (b) The Brillouin zone of the paramagnetic (solid lines) and antiferromagnetic (broken lines) phases of the corundum structure. The band structure in Fig. 8 is shown along the symmetry lines indicated by dot-dashed lines. The $\mathbf{k}$ points $L$, $Z$, and $F$ are on the Brillouin zone of paramagnetic phase and $F/2$ on that of antiferromagnetic phase.

Figure 8

(Color online) (a) Quasiparticle energy bands and (b) the density of states of antiferromagnetic ${\text{V}}_2{\text{O}}_3$. Solid and broken lines refer to those by $\text{U}+\text{GWA}$ and $\text{LSDA}+\text{U}$ with $U=2.6\text{eV}$ and $J=0.9\text{eV}$. The energy zeroth is fixed at the top of the valence band.

Figure 9

(Color online) Imaginary part of the partial Green’s functions of antiferromagnetic ${\text{V}}_2{\text{O}}_3$ by $\text{U}+\text{GWA}$ with $U=2.6\text{eV}$ and $J=0.9\text{eV}$. (a) The total Green’s function (solid line) and experimental XPS spectrum (dotted line) (Ref. 28). (b) The partial Green’s function per atom. The energy zeroth is fixed at the top of the valence band. $a_{1g}$ and $e_g^{\pi }$ originate from $d_{t_{1g}}$ orbitals in the cubic symmetry and $e_g^{\sigma }$ originates from $d_{e_g}$ orbitals.